In practice, the solution \(p({\bf x}),\)  \({\bf x}\in \Omega ,\)  to
Problem REF  might represent the initial distribution of the temperature or the initial stage of the pollutant.
Therefore, computing \(p\)  is important in
many real-world applications, e.g., determination of the spatially distributed temperature inside a solid from the boundary measurement of the heat and heat flux in the time domain [1]};
determining the level of pollutant on the surface of the rivers or lakes [2]};
effective monitoring the heat conduction processes in steel industries, glass and polymer-forming and nuclear power station [3]}.
In the special case when the nonlinear term \(F\)  takes the form \(u(1 - u)\)  (or \(q(u) = u(1 - |u|^{\alpha })\) ) for some \(\alpha > 0\) , the parabolic equation in (REF ) is called the high dimensional version of the well-known Fisher (or Fisher-Kolmogorov) equation [4]}.
Although the nonlinearity \(q\)  does not satisfy condition (REF ), we do not experience any difficulty in numerical computations of the forward problem.
It is worth mentioning that the Fisher equation occurs in ecology, physiology, combustion, crystallization, plasma physics, and in general phase transition problems, see [4]}.
Due to its realistic applications, the problem of determining the initial conditions of parabolic equations has been studied intensively.
The uniqueness of Problem REF  for linear model was proved in [6]}.
The the logarithmic stability results were rigorously proved in [1]}, [3]}.
Due to the presence of the nonlinear term \(F({\bf x}, u, \nabla u)\)  in (REF ), the uniqueness and stability of Problem REF  might need to be proved, especially when the nonlinearity \(F\)  might growth faster than the nonlinearity in [1]}, [3]}.
This task can be done by combining Theorem 1 in [1]} and a truncation technique.
Since the proof is not complicated, we present this result here in this paper for the convenience of the reader.
